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Register a new userGeometric realizations of Lusztigs symmetries
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In this paper, we give geometric realizations of Lusztigs symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztigs symmetries.
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The geometric realizations of Lusztigs symmetries of symmetrizable quantum groups are given in this paper. This construction is a generalization of that in [19].
In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztigs perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztigs symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztigs symmetries on the whole quantum group.
Let $mathbf{U}$ be the quantized enveloping algebra and $dot{mathbf{U}}$ its modified form. Lusztig gives some symmetries on $mathbf{U}$ and $dot{mathbf{U}}$. Since the realization of $mathbf{U}$ by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztigs symmetries on $mathbf{U}$ and their important properties, for instance, the braid relations. In this paper, we define a modified form $dot{mathcal{H}}$ of the Ringel-Hall algebra and realize the Lusztigs symmetries on $dot{mathbf{U}}$ by applying the BGP-reflection functors to $dot{mathcal{H}}$.
Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringels Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [9], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Greens formula.
The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with S. Arkhipov, it relies on technical material developed in a paper with Z. Yun.
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